This paper makes three main contributions. The first main contribution is that we propose a new framework for studying convergence rates of posterior distributions in nonparametric regression, which covers a general class of Gaussian process priors and both the \(L_2\)  and \(L_{\infty }\)  norms. In the existing literature, posterior contraction of nonparametric regression based on GP priors often boils down to the concentration function [1]} to verify the classical prior mass and entropy conditions, which consists of a small ball probability and a concentration measure of the GP prior with certain kernels; for example, see [2]}, [3]} for particular kernels therein. In work along this line, the commonly used metrics for establishing posterior contraction rates are Hellinger and total variation distances. While contraction rates under the \(L_2\)  norm are relatively well studied, there is a limited development on contraction rates under the \(L_\infty \)  norm, with a few exceptions including [4]}, [5]}, [6]}.
